Preheat the broiler. In a 12-inch ovenproof or cast iron skillet, heat the olive oil over medium heat. Add the mushrooms and cook until soft, about 5 minutes. Toss in the asparagus and ham and cook until the asparagus is bright green and barely tender, about 5 minutes. Season with salt and pepper, to taste.

While the vegetables are cooking, in a medium bowl whisk the eggs and thyme. Stir in the goat cheese and salt and pepper, to taste. Reduce the heat to medium-low. Pour the egg mixture into the skillet; do not stir. Cook, undisturbed, until the eggs have set and thickened and only the surface is runny, about 5 minutes.

Put the skillet under the broiler (about 4 inches from the heat) and cook until the center is set and the top is golden, 2 to 3 minutes. Remove the frittata from the oven and let it rest 5 minutes before serving. Loosen the frittata with a spatula and slide onto a plate. Cut in into wedges. Serve the frittata hot or at room temperature.